Let S be a convex, weakly compact subset of a locally convex Hausdorff space ( E , τ ) and f : S → E be a continuous multifunction from its weak topology ω to τ . let ρ be a continuous seminorm on ( E , τ ) and for subsets A , B of E let p ( A , B ) = inf { p ( x − y ) : x  ϵ  A ,  y  ϵ  B } . In this paper, sufficient conditions are developed for the existence of an x  ϵ  S satisfying p ( x , f x ) = p ( f x , S ) . The result is then used to prove several fixed point theorems.